Add the butter to a soup pot and once it melts, drop in the onions, celery and carrots then season with salt and pepper.
Cook over medium heat 2-4 minutes or until the onions are soft then sprinkle the flour over the top. Stir to blend and cook one minute.
Whisk in the chicken stock then add the half and half and broccoli. Cover and cook over low heat 15-18 minutes or until the vegetables are tender.
Turn the heat to low then blend in the cheese and season with salt and pepper. I start with 1/2 teaspoon of salt and 1/4 teaspoon of pepper (add more to your own taste).
The soup is ready to eat at this point but if you like a smoother blend (shown in photos), use an immersion blender and pulse in several areas of the pot until it’s the consistency you like then eat right away.
NOTE: if you don’t have an immersion blender transfer about one cup of the soup to a standard blender, puree then return to the pot.